One-step restarted GMRES stagnation example ($A=\mathrm{diag}(1,-1),b=(1,1{)}^T,x_0=0$)

One step of restarted GMRES searches along the current residual. In the displayed nonsingular system, the first residual minimization is ${\min }_{\alpha }[(1-\alpha {)}^2+(1+\alpha {)}^2]$, whose minimizer is zero. Restarting then repeats the same problem forever and never moves. Unrestarted GMRES instead solves exactly at step two, since $b,Ab$ span the entire space. Thus a smaller search-space cost can come at the price of stagnation; nonsingularity alone does not guarantee progress for one-step restarting.